Source: Besanko & Braeutigam, Microeconomics 5th ed., Chapter 3
Tags: preferences, utility, marginal utility, indifference curves, MRS, marginal rate of substitution, transitivity, completeness, utility function, diminishing marginal utility, perfect substitutes, perfect complements
Difficulty: Intermediate | Prerequisites: Basic supply and demand (Ch. 1-2), comfort with simple algebra and graphing.
This chapter introduces the formal machinery economists use to model how people make choices. It sits right after budget constraints (Ch. 2) and before consumer optimisation (Ch. 4), bridging "what a consumer can afford" with "what a consumer actually picks." You need to be comfortable with the idea that consumers face trade-offs and that we can represent those trade-offs mathematically. If you skipped Chapter 2 on budget sets, go back, because the MRS you learn here plugs directly into the tangency condition you will use in Chapter 4.
Preferences describe how a consumer ranks bundles of goods; utility functions translate those rankings into numbers. The marginal rate of substitution (MRS) tells you the rate at which a consumer is willing to swap one good for another while staying equally happy. Indifference curves are the graphical tool for seeing all of this at once, and their shape changes depending on whether goods are ordinary substitutes, perfect substitutes, or perfect complements.
Preferences
A consumer's ranking over bundles (baskets) of goods, describing which bundles they like more, less, or equally.
Think of it as: your personal ordering of "I'd rather have this than that."
Transitivity
If a consumer prefers basket A to B, and B to C, then they must prefer A to C. One of the core rationality axioms.
In simple terms, this means: your rankings have to be internally consistent. You cannot loop back and prefer C to A after saying A beats B beats C.
Completeness
For any two baskets, the consumer can always say which they prefer, or that they are indifferent. No "I don't know" is allowed.
Think of it as: you always have an opinion when comparing two options.
Utility function
A mathematical function U(x, y) that assigns a number to each bundle such that higher numbers correspond to more-preferred bundles.
In simple terms, this means: a scoring system that respects your preference ranking. The actual number does not matter, only the order.
Marginal utility (MU)
The additional utility gained from consuming one more unit of a good, holding everything else constant. Formally, MUₓ = ∂U/∂x.
Think of it as: how much happier does one more unit make you?
Diminishing marginal utility
As you consume more of a good, each additional unit adds less utility than the previous one. The marginal utility function is decreasing.
In simple terms, this means: the fifth slice of pizza is less exciting than the first.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to give up good Y to get one more unit of good X, staying on the same indifference curve. MRS = MUₓ / MUᵧ.
Think of it as: your personal exchange rate between two goods.
Diminishing MRS
As you move along an indifference curve consuming more of X, you are willing to give up less and less Y for each additional unit of X. This produces the standard convex (bowed-in) indifference curve shape.
Indifference curve
A curve showing all bundles that give the consumer the same level of utility. The consumer is equally happy at every point on the curve.
In simple terms, this means: a contour line on a satisfaction map.
Perfect substitutes
Two goods the consumer is willing to trade at a constant rate regardless of quantities. Indifference curves are straight lines.
Think of it as: sugar and Sweet'N Low if you genuinely do not care which sweetener you use, ounce for ounce.
Perfect complements
Two goods the consumer always wants in a fixed ratio. Extra units of one good without extra units of the other add no utility. Indifference curves are L-shaped.
Think of it as: peanut butter and jelly in a fixed 2:1 ratio. Extra peanut butter without more jelly is pointless.
Economists build consumer theory on three assumptions about how people rank bundles of goods. If any of these fails, the standard model breaks down.
Completeness: the consumer can always compare any two baskets and state a preference or indifference. There is no "I can't decide" option in the model.
Transitivity: if A is preferred to B and B is preferred to C, then A must be preferred to C. A violation would be: preferring A to B, B to C, and then C to A, which creates a cycle with no best option.
More is better (non-satiation): holding everything else equal, more of a good is always at least as good as less. This directly implies that marginal utility is positive (MU > 0).
Transitivity is the one students are most often asked to illustrate with a counterexample. The classic approach: name three baskets A, B, C, state A ≻ B and B ≻ C, then state C ≻ A. That cycle is the violation.
A utility function assigns numbers to bundles so that preferred bundles get higher numbers. The numbers themselves are ordinal: only the ranking matters, not the size of the gap.
U(H) = 10H - H², so MU_H = 10 - 2H.
At H = 2: U = 16, MU = 6
At H = 4: U = 24, MU = 2
At H = 5: U = 25, MU = 0 (utility peaks here)
At H = 6: U = 24, MU = -2 (utility is falling)
At H = 10: U = 0, MU = -10
If hot dogs are free, a rational consumer stops at H = 5, where MU = 0 and utility is at its maximum. Beyond that point, marginal utility goes negative, meaning each extra hot dog makes the consumer worse off. Setting MU = 0 gives: 10 - 2H = 0, so H = 5.
Graphically, the utility function is an inverted parabola peaking at H = 5. The marginal utility function is a downward-sloping straight line crossing zero at H = 5.
For U(x, y) = y√x, the marginal utilities are MUₓ = y/(2√x) and MUᵧ = √x.
Both marginal utilities are positive for any positive x and y, confirming "more is better" for each good.
MUₓ = y/(2√x) is decreasing in x (as x rises, the denominator grows, so MUₓ falls). This is diminishing marginal utility of x.
MUᵧ = √x is constant in y (it does not depend on y at all). The marginal utility of y is not diminishing. Each additional unit of y adds the same amount of utility, √x, regardless of how much y you already have.
The MRS measures the consumer's willingness to trade one good for another along an indifference curve. For two goods, housing (h) and food (f):
MRS(h,f) = MU_h / MU_f
This tells you: "How many units of food would the consumer give up for one more unit of housing, while staying equally satisfied?"
Key points:
If you know MU_h and MU_f at a particular basket, you can calculate the MRS at that basket. You need both marginal utilities.
A diminishing MRS means the indifference curve is convex (bowed toward the origin). As you move right along the curve (more of the horizontal-axis good), the slope flattens because you are less willing to give up the vertical-axis good.
The MRS is the negative of the slope of the indifference curve at any point.
The shape of an indifference curve depends entirely on the relationship between the two goods. You need to recognise each type on sight.
Example: hot dogs and chili, where the consumer likes both.
Curves are smooth, downward-sloping, and convex (bowed toward the origin).
Higher indifference curves (U2 > U1) sit further from the origin (northeast).
The convexity reflects diminishing MRS: as you get more hot dogs, you are willing to give up less and less chili for each additional hot dog.
Example: sugar and Sweet'N Low, accepted ounce-for-ounce.
Curves are straight lines with a constant slope.
The slope equals the constant MRS (here, slope = -1 because 1 oz sugar = 1 oz Sweet'N Low).
Higher indifference curves are parallel lines further from the origin.
Example: peanut butter and jelly consumed in a fixed 2:1 ratio (2 oz PB per 1 oz jelly).
Curves are L-shaped, with the corner (kink) sitting on the line that represents the fixed ratio (here, PB = 2 × Jelly).
Extra peanut butter without more jelly (moving right from the kink) adds zero utility, so the curve is flat horizontal.
Extra jelly without more peanut butter (moving up from the kink) also adds zero utility, so the curve is flat vertical.
Higher indifference curves are larger L-shapes further from the origin.
Example: ice cream (liked) and nuts (neutral, the consumer is indifferent).
Curves are horizontal lines. The consumer's utility depends only on ice cream.
More nuts at the same ice cream level keeps you on the same curve.
Higher curves are higher horizontal lines (more ice cream).
Marginal utility (single good): MU = dU/dG, where G is the quantity of the good.
Marginal utility (two goods): MUₓ = ∂U/∂x (partial derivative holding y constant) MUᵧ = ∂U/∂y (partial derivative holding x constant)
Marginal rate of substitution: MRS(x,y) = MUₓ / MUᵧ = -(slope of the indifference curve)
Utility-maximising consumption of a free good: Set MU = 0 and solve for the quantity. For U(H) = 10H - H²: MU = 10 - 2H = 0, so H* = 5.
Diagram checklist for indifference curves:
Always label axes (good 1 on horizontal, good 2 on vertical).
Mark U1 and U2 with U2 > U1 (further from origin).
Curves must never cross (that would violate transitivity).
Ordinary goods: convex curves. Perfect substitutes: straight lines. Perfect complements: L-shapes with kink on the ratio line.
Diminishing MRS is why people diversify rather than spending all their money on one good. It explains the basic shape of consumer demand: as something becomes relatively cheaper, you buy more of it, but each additional unit matters a bit less.
Perfect complements show up wherever goods must be used together in fixed proportions: left shoes and right shoes, printer cartridges and paper, or a phone plan and a handset on a contract. Perfect substitutes appear when consumers see two goods as interchangeable, such as generic vs. branded paracetamol.
Students often think utility numbers measure "how much" happiness a consumer gets. They do not. Utility is ordinal: it preserves the ranking, but the numerical difference between U = 10 and U = 20 carries no meaning beyond "20 is preferred."
Students confuse diminishing marginal utility with diminishing MRS. They are related but distinct. Diminishing MU is about one good in isolation; diminishing MRS is about the trade-off between two goods along an indifference curve.
A common error on the hot dog problem: setting U(H) = 0 instead of MU = 0. The consumer stops where marginal utility is zero (the peak of the utility curve), not where total utility is zero.
Students sometimes draw indifference curves that cross. This is always wrong: crossing curves violate the transitivity axiom.
The MRS formula (MUₓ / MUᵧ) is heavily tested. Expect to compute it from a given utility function.
Drawing indifference curves for different good types (perfect substitutes, perfect complements, ordinary goods, neutrals) is a staple exam question. Know the shape before you sit down.
The "more is better" implication (positive marginal utility) is a common short-answer or true/false question.
The transitivity violation example is a favourite for early exam questions. Have a clean three-basket example ready.
Knowing when a consumer stops consuming a free good (MU = 0, not U = 0) is a frequent calculation question.
True or False: If U(A) = 15 and U(B) = 30, the consumer gets twice as much satisfaction from B as from A.
False. Utility is ordinal. We can only say B is preferred to A.
Fill in the blank: If preferences are transitive and A ≻ B ≻ C, then A ___ C.
A ≻ C (A is preferred to C).
True or False: The marginal utility of a good can be negative.
True. It can (as with hot dogs beyond H = 5 in the example), but the "more is better" assumption rules this out within the relevant range.
Fill in the blank: The MRS of X for Y equals ___ / ___.
MUₓ / MUᵧ.
True or False: Indifference curves for perfect complements are straight lines.
False. They are L-shaped. Straight lines are perfect substitutes.
Q: Given U(x, y) = x²y, with MUₓ = 2xy and MUᵧ = x², what is the MRS of x for y?
A: MRS = MUₓ / MUᵧ = 2xy / x² = 2y / x.
Q: A consumer has U(H) = 8H - 0.5H². Hot dogs are free. How many does she consume?
A: Set MU = 0: MU = 8 - H = 0, so H = 8. She consumes 8 hot dogs.
Q: Draw the indifference curve shape for a consumer who always uses 3 teaspoons of sugar per cup of tea. What type of goods are these?
A: Perfect complements. The indifference curves are L-shaped, with kinks along the line Sugar = 3 × Tea.
Q: If a consumer's MRS of food for clothing is 3, what does that mean?
A: The consumer is willing to give up 3 units of clothing for one additional unit of food while remaining equally satisfied.
Q: Can two indifference curves from the same preference map ever cross? Explain.
A: No. If they crossed, a bundle at the intersection would lie on both curves, meaning two different utility levels apply to the same bundle. This violates transitivity.
This chapter connects directly to Chapter 4 (consumer optimisation), where the MRS equals the price ratio at the optimal bundle. Without understanding MRS here, the tangency condition in Chapter 4 will not make sense.
Indifference curve shapes come back in Chapter 5 when you study income and substitution effects: the curvature of the indifference curve determines how large each effect is.
The utility function framework also underpins expected utility theory in later courses on risk and uncertainty, where the same "more is better" and diminishing marginal utility concepts reappear in the context of gambles and insurance.
preferences, utility function, marginal utility, MU, diminishing marginal utility, marginal rate of substitution, MRS, indifference curve, indifference map, ordinal utility, cardinal utility, transitivity, completeness, non-satiation, more is better, convex preferences, perfect substitutes, perfect complements, L-shaped indifference curves, linear indifference curves, Besanko Braeutigam Chapter 3, intermediate microeconomics preferences, consumer theory, utility maximisation, rate of substitution, trade-off rate, consumer choice, preference axioms, rational preferences